Cutting a 2D plane using intersecting planes as a guide
As it is a "cut" question, you can use knife project :


In object mode select the "wall" part RMB
Then both select the "floor" part Shift+RMB
Enter edit mode Tab
Place the view to the top ortho NumPad7
Then use knife project in the tool panel
Last step : remove the unwanted vertex




Knife project uses a mesh to cut into another. The cuts are given through the view plane (so it depends on from where you are looking). The cuts are given using the cutting object edges (from this point of view), to create new vertices and edges in the cut mesh.

A step further... this make an ngon (a face with more than 4 vertices) and it is better to avoid ngons...

First approach :


Enter edit mode
Select all and Ctrl+T to triangulate
Then selecting quad parts Alt+J


 

Second approach :


Remove the ngon face
Subdivide the wanted edges W then subdivide
Place the 3D cursor at the 'pivot vertex'
Set pivot mode on 3D cursor
Then move (scale) the vertices to the 3D cursor
Last : remake the faces F and eventually correct the normals Ctrl+N




Other possible approaches, certainly...